Asymmetric Plasmonic Nanoshells as Subwavelength Directional

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Asymmetric Plasmonic Nanoshells as Subwavelength Directional Nanoantennas and Color Nanorouters: A Multipole Interference Approach Giovanni Pellegrini,* Paolo Mazzoldi, and Giovanni Mattei Department of Physics and Astronomy, University of Padova and CNISM, Via Marzolo 8, 35131 Padova, Italy S Supporting Information *

ABSTRACT: Asymmetric plasmonic nanoshells (ANs), consisting of an off-center dielectric core and a metallic shell, are proposed as subwavelength directional nanoantennas, when coupled to a localized source of electromagnetic radiation. Unidirectional emission and color routing are obtained at multiple wavelengths and for different subwavelength AN geometries and sizes. ANs are capable of 2 orders of magnitude directional emission enhancements and of forward−backward directional gains as large as 30 dB. The obtained results are straightforwardly interpreted in terms of interference phenomena in coherent multipole sums.

M

region.29,30 We employ an extension of Mie theory to show that ANs are able to support combinations of plasmon modes capable of redirecting the radiation of a local source. Directional emission properties are then analyzed as a function of the geometrical and spectral characteristics of the nanostructures. We finally show by straightforward Mie theory that the obtained results are easily interpreted in terms of interference phenomena in coherent multipole sums. Figure 1 reports a schematic description of the AN configuration. The dielectric core and metallic shell sizes are defined by their respective radii Rc and Rs. The dielectric core position is fixed in the origin of the reference frame, and the metallic shell is displaced by an offset d in the z axis negative direction. The minimum shell thickness t is defined as t = Rs − Rc − d. In a first scenario, ANs are illuminated by x polarized plane waves traveling in the positive and negative z axis directions and labeled z+ and z−, respectively. Otherwise ANs are illuminated by local sources defined as classical electric dipoles located along the z axis in the origin (center position), at the cavity edge (top position), and outside the NA (off position). In this context, dipoles are modeled as pure sources of the incident field; that is, they are not driven by the field of the other dipoles and by the field scattered by the antennas. This corresponds to an experimental setup where a Stokes shift between excitation and emission exists.14,15 Dipole moment orientations are along the x and z axes and labeled as m = ±1 and m = 0, respectively: this reflects the convention that n,m are the multipolar indexes; therefore, according to the azimuthal symmetry of the structure, m = 0 refers to plasmons polarized

etallic nanostructures have the ability to modify and enhance the electromagnetic radiation emission of localized sources, and conversely they are able to focus light at the nanoscale.1−6 This phenomenon is largely analogous to what happens in classical antennas (hence the nanoantenna (NA) definition) and is more pronounced if the radiation frequency matches the localized surface plasmon resonance (LSPR).7,8 A variety of geometries including nanorods, nanodisks, particle multimers, bow-tie, and regular array NAs have been studied to obtain different effects such as local-field and emission enhancement, polarization tailoring, and angular emission redistribution.8−13,52 While enhancement effects have been the first to attract considerable attention, recently a substantial amount of work has been devoted to understand, tailor, and modify NA directional emission properties. Large directivities have been demonstrated, either theoretically or experimentally, in a wealth of different nanostructures, including nanoparticle and nanohole arrays,14−19 patch and nanowire NAs,13,20 Yagi-Uda configurations,21−25 bulls eye designs, and nanorod and nanodisk compact multimers.8,26−28 Yet, these arrangements are usually large if compared with the radiation wavelength or, in the case of subwavelength structures, base their directivity properties on the interaction of multiple units and on precise placement of the local source.8,27,28 In this paper, asymmetric plasmonic nanoshells (ANs) are proposed as subwavelength directional nanoantennas, when coupled to a localized source of electromagnetic radiation.29−35 ANs consist of a dielectric core surrounded by a metallic shell, where the core is offset with respect to the shell center. We aim to obtain directional emission by exploiting the shell asymmetry and consequently the strong coupling between dipolar and higher-order modes supported by the NA in the same spectral © 2012 American Chemical Society

Received: June 19, 2012 Revised: September 13, 2012 Published: September 14, 2012 21536

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σsca =

4π k2

∞

n

∑ ∑

(|anm|2 + |bnm|2 )

(3)

n = 1 m =−n 37

where only coefficient amplitudes are involved. Localized light sources are described in our theoretical framework as o classical dipoles with intrinsic γro and γnr radiative and nonradiative recombination rates. The influence of the AN on the radiation processes is easily expressed in terms of power dissipated inside the nanoantenna and radiated by the nanoantenna emitter ensemble. Normalized modified rates are then expressed as γr/γor = Pr/Por and γabs/γor = Por , with γabs being the contribution to nonradiative recombination rates related to the power dissipated inside the nanoantenna, Pabs, Pr the power radiated by the emitter−antenna system, and Por the power radiated by a classical dipole.40−43 The overall o nonradiative rate may be finally written as γnr = γnr + 12,43,44 γabs. In the case of a dipole emitter embedded in the AN dielectric core, the explicit expression for the modified radiative recombination rate is equal to that of the scattering cross-section in eq 3, excluding a multiplying constant. The configuration with an emitter external to the AN requires instead the substitutions anm → anm + pnm and bnm → bnm + qnm, where the dipole pnm and qnm expansion coefficients, calculated in the shell reference frame, are added to take into account the coherent superposition of incident and scattered electromagnetic fields.45 The first studied AN consists of a SiO2, Rc = 50 nm dielectric core and an Ag, Rs = 75 nm metallic shell surrounded by water (n = 1.33), where all the relevant optical constants are retrieved from the literature in tabulated form.46 The core−shell offset is fixed at d = 20 nm, with the minimum shell thickness left to be t = Rs − Rc − d = 5 nm. Figure 2a shows scattering spectra normalized to unity along with their multipolar decomposition for the above-described AN illuminated by z+ and z− plane waves as defined in Figure 1. In the adopted theoretical framework, the scattering cross-section is defined as in eq 3, and likewise a single multipolar contribution may be written as 2 2 (4π/k2)∑m=n m=−n|anm| + |bnm| . A broad dipolar resonance is clearly visible at λ ∼ 680 nm for both z+ and z− illumination. The resonance shape is similar in each case, especially on the long wavelength size, in spite of the off-center inclusion. The nanoshell asymmetry is instead reflected in the scattering spectra at around λ ∼ 520 nm, where in the case of z+ illumination a more pronounced dip is apparent. The asymmetry emerges more sharply in the spectral multipolar decomposition, which highlights a strong quadrupolar component at λ ∼ 520 nm, while the dipolar contribution is attenuated in the same spectral region. This spectral feature may be straightforwardly interpreted as a Fano resonance signature, where the AN symmetry breaking allows the nondiagonal coupling between a broad dipolar mode and a sharp quadrupolar one. The asymmetry in the scattering spectrum continues on the short wavelength side, where both the overall shape and the underlying multipolar components show sizable differences. The asymmetries highlighted in the scattering quantities are also markedly visible in the absorption spectra reported in Figure 2b. Prominent spectral features emerge on the short wavelength side of the spectrum with sharp absorption peaks arising in correspondence of the scattering spectral dips, as a clear indication of local-field enhancement at the level of the metallic shell. The weight of the multipole contributions roughly follows the behavior observed for the scattering ones, with the notable exception

Figure 1. (a) Sketch of typical AN configuration. Rc,Rs, core and shell radii; d, core−shell offset; t = Rs − Rc − d, minimum shell thickness; z+, z−, x-polarized incident plane waves. Dipole sources are indicated by the red arrows and placed at the origin (center position), at the cavity edge (top position), and above the NA (of f position). Dipole moment orientations are along the x and z axes and labeled as m = ±1 and m = 0, respectively.

along the polar axis, and m = ±1 refers to plasmons polarized perpendicular to the polar axis.29 A sample experimental system close to our theoretical model could be constituted by ANs synthesized by colloidal or sputtering techniques,30,36 with emitters in the form of molecules or quantum dots either deposited on the outer shell or embedded in the dielectric core. Our theoretical framework follows the generalized multiparticle Mie (GMM) approach introduced by Xu37 and is adapted to the analysis of ANs along the guidelines proposed by Borghese:38,39 more specifically, vector translation theorems (VTT) are employed to bring the off-center inclusion and the outer shell to a common reference frame, to easily solve the Maxwell equations. Moreover, the modeling framework has been extended to included electric dipoles as sources of the incident field. In the listed cases of plane wave and dipole illumination, transverse far-field scattering components are written in spherical coordinates as ⎡ Esθ ⎤ ⎢ ⎥= ⎢⎣ Esϕ ⎥⎦

⎛⎡ ia ξ ′ ⎤ ⎡ bnmξn ⎤ ⎞ nm n ⎥τnm − ⎢ ⎥πnm⎟eimϕ Enm⎜⎜⎢ ⎟ ⎢ ⎥ ⎢ ′ ib a − ξ ξ ⎣ ⎦ ⎣ ⎦ ⎠ ⎝ nm n ⎥ nm n n = 1 m =−n ∞

n

∑ ∑

(1)

and ⎡ Hsθ ⎤ k ⎢ ⎥= ⎢⎣ Hsϕ ⎥⎦ ωμ0

⎛⎡ ia ξ ⎤ ⎡ b ξ′ ⎤ ⎞ nm n ⎥πnm + ⎢ nm n ⎥τnm⎟ Enm⎜⎜⎢ ⎢⎣−anmξn ⎥⎦ ⎟⎠ n = 1 m =−n ⎝⎢⎣ ibnmξn′ ⎥⎦ ∞

n

∑ ∑

(2) eimϕ where anm and bnm are the electric and magnetic scattering coefficients; ξn, ξ′n are radial Riccati−Bessel functions and their derivatives with respect to their argument; πnm and τnm are angular functions defined in terms of associated Legendre polynomials; and Enm are auxiliary normalization coefficients.37 The remaining terms have their usual meanings; that is, k and ω correspond to the light wavevector and angular frequency, while μ0 stands for the vacuum permeability. In this framework, the scattering cross-section expression for a plane wave illuminated AN may be easily obtained by integration of the Poynting vector over a surface enclosing the antenna and eventually written as 21537

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can be hardly assigned to a specific multipolar mode, since dipole and quadrupole amplitudes have both significant intensities at the analyzed wavelength. The largest enhancements are seen for z− illumination, with field localization in the middle of the spherical cavity, as it distinctly stems from Figure 2c, indicating that, in terms of reciprocity, ANs are able to support unidirectional radiation even at shorter wavelengths. Recombination rate enhancement and the relative multipolar decomposition spectra for dipolar emitters located in the AN cavity are illustrated in Figure 3a, where the multipolar decomposition contributions are proportional to (4π/ 2 2 k2)∑m=n m=−n|anm| + |bnm| as for the plane wave illumination. In

Figure 2. Rs = 75 nm AN: (a,b) z+ and z− scattering and absorption spectra and multipole contributions; (c) z+ and z− z axis local-field profiles; (d−g) z+ and z− local-field mappings on the xz plane. In (a,b) dots mark the mapping wavelengths; in (c) dots mark the emitter positions (see text for further explanations).

that absorption components may assume negative values. This at first counterintuitive result is easily explained if we remind that multipole contributions of the absorption cross section are obtained by integrating the incoming and outgoing field components at the outer shell surface. Symmetry breaking allows nondiagonal coupling between waves of different order, therefore introducing an unbalance in the single multipole contributions, while of course the overall absorption cross section remains positive. Figure 2d,e reports local-field enhancement mappings for z+ and z− illuminations at λ ∼ 520 nm. The already discussed structural and spectral asymmetries are reflected and amplified in the local-field distribution: the z− plane wave at 520 nm returns a field pattern that is mainly dipolar in nature, with a nearly uniform distribution inside the cavity, while z+ illumination does not show any particular multipolar symmetry given the strongly hybridized nature of the involved plasmon mode, with dipole and quadrupole contributions having the same order of magnitude.29,30 A large |E| ∼ 16 local-field hot spot, localized on both the cavity and the external surface side, is observed at the level of the shell thinning. Its spatial localization is better highlighted in Figure 2c, which reports field profile plots along the z axis: at z = 40 nm, z+ local-field enhancements are found to be one order of magnitude larger than in the z − configuration, a result that in view of the reciprocity principle suggests AN as an ideal candidate to support strong unidirectional radiation phenomena.8,47 Examples of z+ and z− local-field distributions for the short wavelength side of the spectrum (λ = 380 nm) are reported in Figure 2f,g. The asymmetry emerging in the absorption and scattering quantities is visible in the local-field maps, though the field distributions

Figure 3. Rs = 75 nm AN: (a) radiative rate enhancement and multipole spectra for top and center m = ±1 emitters; (b) Δϕ12 dipole−quadrupole phase relationships for top and center m = ±1 emitters; (c) directional gain (eq 4) polar plots for emitters at the center, top, and of f position at the respective resonance wavelengths, as indicated in (a). 21538

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the modeled configuration, the Hertzian dipoles representing the emitters are located in the center (axes origin) and top (10 nm below the cavity edge) position illustrated in Figure 1 and oriented along the x axis. The placement of the emitters appears logical in view of the field distributions shown in Figure 2d,e; the chosen positions allow us to weight the contribution of the higher-order multipoles introduced by the symmetry breaking and to probe the interaction between an emitter and the hybridized Fano resonance supported at 520 nm. It is immediate to see that the maximum radiative enhancement for the top local source is found at 520 nm, in correspondence to the spectral Fano dip for z+ illumination, thus highlighting the strong coupling between the emitter and the hybridized dipole−quadrupole mode. The analysis of the multipole composition reveals that the dipolar and quadrupolar amplitudes equally contribute to the radiation enhancement peak, with the multipolar terms presenting comparable amplitudes in a spectral band of tens of nanometers around the Fano resonance wavelength. A final element of insight is provided by the observation of the multipole coefficient phase relationship Δϕ12, as illustrated in Figure 3b. The dephasing changes slowly as a function of the wavelength and exhibits a near perfect phase opposition at the Fano resonance position, therefore suggesting that a precise phase relationship between multipolar components may play a role in terms of the AN directional properties. The radiation enhancement spectra for center emitters are reported in the lower part of Figure 3a. The spectral shape displays a more complex appearance than in the former case, with two enhancement peaks at about 550 and 360 nm. The nature of the long wavelength maximum may be explained if the multipolar decomposition of Figure 3a and the local-field mappings of Figure 2d,e are properly considered: the quadrupole contribution to the radiative rate enhancement clearly presents a peak at λ ∼ 520 nm, therefore meaning that the dipole in the center of the cavity can couple successfully to the Fano mode yet less efficiently than for the top configuration since the mode is strongly localized at the level of the metallic shell thinning. The mentioned local-field profiles instead reveal that, even if the Fano mode is still dominant in the origin at the considered wavelength, the dipolar mode excited by z− illumination has a lower yet comparable strength in the cavity center, so adding a strong dipolar component to the emission rate multipolar decomposition. A similar but simpler analysis can be performed for the 360 nm radiative maximum. In this case, the field mappings of Figure 2f,g and the field profiles of Figure 2c emphasize the fact that the emitter is strongly coupled to the dipole mode excited in the z− configuration. The multipolar decomposition confirms this interpretation, reporting only a minor quadrupolar contribution at the radiation short wavelength maximum, attributed to the rapid spatial variation of the incident field. The dipole−quadrupole phase relationship, illustrated in the lower portion of Figure 3b, presents a near phase opposition at 520 nm, while at 360 nm the dipole and quadrupole contribution may be roughly considered to be in phase. To verify the presently introduced conjectures about the AN directional radiation properties, we performed directional gain polar plots for all the emitter configurations (center, top, of f) and emitter dipole moments along the x (m = ±1) and z (m = 0) axes, as illustrated in Figure 3c, where the directional gain is defined as8,15

D(θ , ϕ) = 4π

S(θ , ϕ) 2π

π

∫0 ∫0 S(θ , ϕ)sin θ dθ dϕ

(4)

with S(θ,ϕ) being the far-field Poynting vector module. Focusing on the x oriented emitters, the results presented in the two top panels of Figure 3c clearly show that the radiation emitted by the dipoles at λ ∼ 520 nm, either in the top or the of f position (10 nm below or above the internal and external shell edges), couples to the Fano mode and is strongly modulated toward the θ = π downward direction. The interpretation of the result is straightforward considering the local-field profiles of Figure 2c, since reciprocity dictates equal receiving and emitting patterns for a generic plasmonic nanoantenna. The study of the center configuration directional radiation properties requires instead the introduction of two directional gain polar plots, one for each of the radiative enhancement maxima, as illustrated in the bottom panels of Figure 3c. It is immediately apparent that the main feature of the center configuration is that the dipole radiation is routed in opposite directions for different wavelengths, with upward θ = 0 radiation for the shorter wavelength resonance and downward θ = π emission for the longer wavelength one. The present results are again consistent with the reciprocity theorem interpretation, if the field profiles of Figure 2c are taken into account: at 520 nm, local fields at the cavity center are dominated by modes excited in z+ illumination, while the opposite is true at 380 nm. Similar considerations may be expressed when dealing with the m = 0, z oriented case, when some relevant differences are taken into account. Without entering the details of the discussion, we note that the polar symmetry of the structure naturally constrains the radiation pattern symmetry and shape, with no radiation lobes along the z axis. Therefore, while unidirectional hemispherical emission is attainable even in the m = 0 case, the radiation will not be focused on a single principal scattering lobe but instead on an emission cone revolving around the z axis. It is now interesting to introduce the analysis of a larger AN capable of supporting higher-order multipoles, to study the different emerging directional radiation properties. We choose a SiO2, Rc = 160 nm dielectric core and a Ag, Rs = 200 nm metallic shell in an n = 1.0 matrix. The core−shell offset is d = 30 nm for a final minimum shell thickness of t = 10 nm. It is immediately evident from a first glance to Figure 4a that the scattering spectra and multipolar decompositions emerging from z+ and z− illumination present a rich and complex organization. The only component susceptible of a straightforward interpretation, as well as the only part of the spectra symmetric for z+ and z− incident plane waves, is the broad dipolar contribution located at about λ ∼ 1100 nm. The remaining portion of the scattering spectra displays instead an intricate interplay between all the multipolar components and a marked asymmetry for the two illumination configurations throughout all the visible range. The multipolar analysis finally emphasizes a sizable octupolar contribution toward the blue end of the spectrum, in agreement with what is expected in the case of a larger AN. One more quick look at the spectra reveals that they are characterized by numerous scattering dips, in contrast to the former case where only one was found. Starting from the long wavelength side of the spectrum, at least three dips can be identified for z+ illumination, at 680, 570, and 480 nm, respectively. As before, the multipole decomposition analysis is helpful in revealing the physical mechanisms at the 21539

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emerge between the two structures, and in particular, in the case of the larger AN, a much weaker field localization is noticed for z+ incident light, along with a predominance of the dipolar cavity mode excited by z− illumination, as is also immediately apparent from Figure 4c. This last observation is in line with the 2-fold nature of the resonance just described above that is a superposition of cavity and coupled multipole modes. A similar discussion may be considered for the 570 nm resonance, whose field distribution displays a striking similarity with the one found for the smaller AN 520 nm resonance. Upward illumination causes large field enhancements strongly localized at the shell thinning, while a nearly uniform field distribution is found inside the cavity for incident light in the opposite direction. The different nature of the present resonance, that is, the quadrupole−octupole coupling, stems from the more structured and complex appearance of the local fields: it is indeed possible to notice that additional field hot spots are present at the nanoshell equatorial level (Figure 4f) and that additional field maxima and minima become visible in the field profiles, at z ∼ −80 nm and z ∼ 20 nm in the case of z+ illumination and at z ∼ −50 nm and z ∼ 80 nm in the opposite situation. The 400 nm spectral dip shares a common origin with the just-analyzed resonances, yet it exhibits the major difference of being excited by a downward incident plane wave. This new element allows us to speculate about the localfield enhancement localization, that in the present case should be centered on the opposite AN side, that is, at the level of the thicker shell. Our conjecture is fully confirmed by the local-field mapping displayed in Figure 4i, where it may be clearly seen that the local field focuses over the thick side of the plasmonic shell, in a specular fashion with respect to the former field configurations. A reading of the local-field profiles of Figure 4c in terms of the reciprocity principle allows us to speculate about which resonances may be susceptible to radiation unidirectional routing, if an emitter is placed inside the cavity. In the case of emitters in the top position, close to the shell thinning, it is easy to anticipate that only the 570 and 400 nm resonances would be able to sustain unidirectional radiation in the downward and upward direction, respectively, while at 680 nm, the local-field profiles reveal equal field enhancements at the top position for both upward and downward illumination, therefore excluding any possibility of unidirectional radiation patterns. A simpler examination may be put forward for the center configuration since emitters in the center of the cavity would obviously couple only to the longer-wavelength 680 nm mode, given the local-field minima featured in the origin by all the other resonances. The local-field profiles of Figure 4c indeed show that the center configuration would in principle be able to support directional enhancement, even if a weaker field asymmetry would lead to a smaller effect than the one expected for emitters in the top position. If now the attention is directed toward the AN rate enhancement properties, the upper part of Figure 5a clearly illustrates that emitters in the top position (20 nm below the cavity internal surface) undergo a considerable rate enhancement in correspondence of all the spectral scattering dips. Starting to investigate the resonances in order of decreasing wavelength, the 700 nm maximum immediately reveals nearly equal dipolar and quadrupolar contributions originating from the AN symmetry breaking and a slight shift to the red if compared with the corresponding scattering spectral feature, explained in terms of the red-shifted quadrupole amplitude in the z+ scattering spectrum. Similar considerations may be

Figure 4. Rs = 200 nm AN: (a,b) z+ and z− scattering and absorption spectra and multipole contributions; (c) z+ and z− z-axis local-field profiles; (d−i) z+ and z− local-field mappings on the xz plane. In (a,b) dots mark the mapping wavelengths; in (c) dots mark the emitter positions (see text for further explanations).

origin of these spectral features, which in the present situation possess a more complex nature than observed before. The arising of the spectral dips can indeed be assigned to two mechanisms of different nature: the first mechanism stems from an interference between the incident and scattered field which leads to a resonant suppression of the scattering and to cavity modes featuring large field enhancements.48 The second mechanism instead, as highlighted before, originates from the off-diagonal multipole−multipole interactions. In the case of the 680 nm resonance, the coupling mainly involves dipolar and quadrupolar modes, while in both of the shorter wavelength dips the interaction is largely dominated by quadrupolar and octupolar modes. The same analysis may be applied in the case of z− illumination, having care to note the arising of new resonances (400 nm), the vanishing of some resonances present in the former case (570 nm), and the different weight of the multipolar contributions in each of the involved spectral features. The analysis of the absorption spectra of Figure 4b confirms the conclusions drawn by the observation of the scattering cross sections. As seen in the case of the smaller Rs = 75 nm AN, marked absorption peaks emerge in correspondence of the scattering spectral dips, while the hierarchy of the multipole amplitudes among the scattering and absorption multipolar decompositions is approximately preserved. The examination of the local-field profiles and mappings of Figure 4c−i may help to shed further light on the nature of the above scattering and absorption resonances. Field mappings for the 680 nm spectral dip (Figure 4d,e) promptly reveal an intensity distribution largely analogous to that found for the smaller AN at 520 nm, in line with the similar weight of the multipolar contributions. Nevertheless, notable differences 21540

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phase, as was roughly the case for the Rs = 75 nm shell, while at 570 nm the dominating multipoles are in nearly perfect phase opposition, bearing again a strong similarity with what is seen in the first analyzed structure. As usual, the center emitters display a phase relationship much closer to a quadrature configuration, again showing an affinity with the Rs = 75 nm AN. Since multipole amplitudes and phase relationships at the analyzed wavelengths bring a close resemblance to those of the former structure, and recalling the local-field results in terms of reciprocity, it is reasonable to expect, even in this case, the arising of directional radiation properties. The above speculations concerning the directional properties of the considered AN are readily verified by analyzing the directional gain polar plots reported in Figure 5c, first for the x (m = ±1) oriented emitters. The results illustrated in the upper two panels clearly show that emitters coupling to the 570 nm resonance, either in the top or off position (20 nm below or above the shell surfaces), see their radiation mainly routed toward the downward θ = π direction in agreement with the evidence emerging from the local-field plots and reciprocity considerations. The radiation pattern for top dipoles coupled to the 400 nm resonance, reported in the lower left panel of Figure 5c, displays instead a strong θ = 0 radiation lobe, therefore opening the chance for directional color routing even in the case of larger geometries. Finally, in the case of center dipoles coupled to the 680 nm resonance, a weaker yet sizable unidirectional radiation pattern is observed, again in agreement with the consideration emerging from the analysis of the weaker field asymmetry of Figure 4c, interpreted in the light of the reciprocity theorem. If z (m = 0) oriented emitters are considered, it may be seen that hemispherical unidirectional radiation is obtained for top and off position at 570 nm, with the obvious additional constraints imposed by the polar symmetry of the structures. No unidirectional radiation is instead observed in the remaining cases, possibly because of the larger size of the nanoantenna and the arising retardation effects. Since we are interested in the AN unidirectional emission properties, it finally appears logical to analyze the spectral properties of the forward and backward emission enhancement factors, defined, respectively, as S+ = SAN(0,0)/Sdip(0,0) and S− = SAN(π,0)/Sdip(π,0), with SAN and Sdip being the Poynting vector modules of the AN and isolated dipole, respectively. Given that the observable quantities introduced so far are unable to represent the AN unidirectionality, we shall also monitor the forward (θ = 0) to backward (θ = π) (FB) radiation ratio, defined in dB as G± = S+,dB − S−,dB = 10 log10(S+/S−), where S+,dB = 10 log10(S+) and S−,dB = 10 log10(S−), respectively. In view of the reciprocity principle, it is also expected that a nanoantenna should have very similar reception and transmission patterns, thus to the check the validity of our calculations we defined the forward to backward reception ratio as K± = log10(|Ep+|2/|Ep−|2), where |Ep+|2 and |Ep−|2 are the field amplitudes for both plane wave illumination directions and p stands for the different dipole positions. The upper part of Figure 6a reports transmission and reception FB gain ratios for the Rs = 75 nm AN, for all three emitter positions, with dipole moments oriented along the x axis (m = ±1). Large positive values imply strong upward emission, while large negative values have the same meaning for the downward radiation. The first remarkable feature is that G± and K± are indistinguishable, therefore validating the reciprocity principle for the structure in question. More importantly, extremely large

Figure 5. Rs = 200 nm AN: (a) radiative rate enhancement and multipole spectra for top and center m = ±1 emitters; (b) Δϕ12,23 dipole−quadrupole and quadrupole−octupole phase relationships for top and center m = ±1 emitters; (c) directional gain (eq 4) polar plots for emitters at center, top, and of f position at the respective resonance wavelengths, as indicated in (a).

invoked for the 570 nm resonance, which of course displays a quadrupole−octupole nature in agreement with what is seen in the scattering and absorption spectrum. Directing now the analysis on the 400 nm maximum, it is remarkable to note that the emitter in the top configuration is able to couple to the mode even if the local-field enhancement is located on the opposite shell edge. Nevertheless, the coupling happens by means of a secondary field enhancement hot spot located at z ∼ 100 nm, which allows the efficient out-coupling of the dipole radiation. The multipole phase relationships reported in Figure 5b display an intricate behavior; nevertheless, it is possible to try a comparison with what is observed for the smaller AN. In the case of the top emitters for the short wavelength 400 nm rate enhancement maximum, all the multipoles are nearly in 21541

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such anomalous FB peaks, so to attribute a large forward G± gain to an extremely small backward S−,dB term and vice versa. To achieve a deeper understanding of the AN directional radiation properties, we introduce a simple theoretical model which aims at explaining the obtained radiation patterns in terms of interference phenomena occurring in coherent multipole sums. More precisely, we study the angular dependence of S(θ,ϕ) in the framework of Mie theory, turning on only specific multipole combinations, with well-defined amplitude and phase relationships. Since the total scattering cross-section of a nanoshell is only a function of amplitudes (see eq 3), it is clear that interference phenomena between multipoles of different order can not occur.49 In the case of multisurface particles in general, and for AN in particular, interferences may obviously be observed due to the coupling between different plasmon modes supported by the structure, but these effects are not a direct consequence of the functional form of the cross-section. Interference phenomena can instead be observed in differential scattering quantities and likewise in 1 the scattering Poynting vector Ss = 2 9{Es × Hs}where Es and Hs are, respectively, the electric and magnetic scattering fields and 9 stands for the real part of the expression.49,50 When dealing with scattering quantities, the Poynting vector angular dependence is best expressed in far-field asymptotic form: the radial component equals the vector modulus, which is written in spherical coordinates as 1 * * |Ss| = Ss,r = − 2 9{Es, θ Hs, ϕ − Es, ϕHs, θ}. The asterisk stands for complex conjugation, and Es,θ,ϕ,Hs,θ,ϕ are the transverse components of the scattered field in spherical coordinates. Explicit expressions for Es,θHs,ϕ * and Es,ϕHs,θ * are easily obtained from the field components of eq 1 and eq 2 and clearly show that scattering patterns are potentially subject to interference phenomena between multipoles of different orders. Since we are mostly interested in plasmonic particles smaller or comparable to the wavelength of interest, and to preserve the model simplicity, magnetic coefficients bnm are neglected in the following model as a first approximation, and only electric anm coefficients are retained. The polar symmetry of structures additionally dictates that only the m = 0 and m = ±1 coefficients are nonzero, leaving us with analysis of scattering quantities depending only on an,m=0 and an,m=± 1 pure multipoles.29 Furthermore, scattering pattern shapes are only a function of multipole coefficients relative amplitude and phase; therefore, in our model a reference an=1,m coefficient may be set to unity, and the other values may be then assigned accordingly. Figure 7a shows directional gain polar plots for a1,0 = 1 and a1,−1 = −a1,1 = 1 coefficients, that is, for dipoles oriented along the z and x axis, respectively. This leads to the usual ″donut-shape″ radiation patterns, which may result from a plane-wave illumination of a subwavelength particle or from a feeding dipolar local source placed in the center of a symmetric nanoshell antenna. Radiation patterns corresponding to a2,0 = 1, a2,−1 = −a2,1 = 1 (quadrupole) and a3,0 = 1, a3,−1 = −a3,1 = 1 (octupole) multipoles are reported in Figure 7b,c. It is clear from a first look that the number of the radiation lobes increases as a result of increasing mode order, while on the contrary the beam-width of each lobe decreases as expected. Multipole directivity, defined as D = Max[D(θ,ϕ)], may be easily seen to be a linearly increasing function of the multipolar order n, and in particular we have Dn = (2n + 1)/2,51 in good agreement with our directional gain polar plots.

Figure 6. (a) Directional forward (θ = 0) and backward (θ = π) G± and K± spectra for both AN and for all emitter positions. Cross markers illustrate a subset of K± spectral values: G± and K± would be indistinguishable at the plot scale; (b−c) forward S+ and backward S− emission enhancement spectra for both AN and for all emitter positions. Dots mark the resonance wavelengths, where local-field and gain polar plots are performed.

FB directional gains, between 20dB and 30dB, are found for emitters in the top and off configurations, in correspondence of the backward emission enhancement maxima (S− ∼ 100, upper Figure 6b), with FB gain comparable with the best performances of subwavelength structures with similar size.8 Finally, lower G± ∼ 10dB gains as well as S+,− ∼ 5−10 emission enhancements are seen for emitters in the center configuration, which are nevertheless remarkable values in view of the reduced size of the structure and of the obtained directional color routing. Similar considerations may be expressed for the emission enhancement and FB gain properties of the Rs = 200 AN, yet smaller emission enhancements (S+,− ∼ 10) and FB gains (G± ∼ 10dB) are found for all the emitter positions at the principal resonance wavelengths. We eventually underline that additional care must be taken when interpreting the G± data, since large G ± ∼ 20dB minima and maxima appear in the spectrum, in the absence of a corresponding forward or backward emission enhancement peak. The results are easily explained if it is noticed that nearly complete suppression of forward or backward radiation occurs in correspondence of 21542

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Figure 7. Directional gain (eq 4) polar plots for pure (a) dipole, (b) quadrupole, and (c) octupole an,m=−1,0,1 scattering coefficients. Plus and minus symbols indicate the phase of each scattering lobe.

It is now straightforward to understand that directional radiation patterns may be obtained by exploiting the coherent multipole sum and the phase relationships between the different scattering lobes. To better investigate this matter, it is convenient to initially discuss the coherent superposition of a dipole and a quadrupole, in the m = ±1 case. The phase relationship between the upward (θ = 0) and downward (θ = π) dipolar scattering lobes is easily determined if we recall that the expression for the electric radiation field includes the term (r × p) × r, where the obvious consequence of a π dephasing for the radiation moving in the opposite directions originates. A similar relationship can be demonstrated for the quadrupolar scattering, where phase opposition roughly holds between consecutive radiation lobes.51 The present rule of thumb can be further generalized to higher-order multipoles, with neighboring lobes approximately in opposition of phase and phase matching for every other lobe.51 These simple qualitative considerations allow us to anticipate that an in phase dipole− quadrupole combination (Δϕ12 = 0) will be characterized by upward constructive and downward destructive interference, and the opposite would happen for a phase opposition (Δϕ12 = π) combination, thus providing a straightforward mechanism to obtain strong directional radiation. Figure 8a shows polar radiation plots for unitary amplitude dipole−quadrupole coherent sums, where a1,−1 = −a1,1 = 1, a2,−1 = −a2,1 = eiΔϕ12 and Δϕ12 assumes 0, π/2, and π values. It is clear from a first analysis that the above qualitative discussion is indeed accurate and that constructive and destructive interferences operate to produce unidirectional radiation patterns. The two different cases of in phase (Δϕ12 = 0) and phase opposition (Δϕ12 = π) sums display the expected behavior, with strong upward and downward unidirectional radiation, respectively. Side scattering is partially suppressed and slightly bent along the direction opposite to the principal radiation lobe: this effect can again be explained in the interference picture introduced before since the quadrupole side radiation will interfere destructively with the upper dipolar radiation lobe and constructively with the lower one. When the dipole and the quadrupole are combined in quadrature (Δϕ12 = π/2), no unidirectional emission is obtained, in reason of the fact that the phase relationship between upper and lower scattering is identical, therefore leading to a symmetric pattern. The next natural step is to focus the attention on the coherent

Figure 8. Directional gain (eq 4) polar plots for (a) dipole− quadrupole, (b) quadrupole−octupole and (c) dipole−quadrupole− octupole coherent sums. Δϕ12 and Δϕ23 indicate the dipole− quadrupole and quadrupole−octupole phase relationships.

quadrupole−octupole combination, again for the m = ±1 case, as shown in Figure 8b. The description of the dipole− quadrupole sum can be readily transposed to the present case, provided that some minor differences are considered. As seen before, destructive and constructive interferences along the z axis direction give rise to unidirectional upward emission for the in phase coherent sum (Δϕ23 = 0) and to the opposite situation if phase opposition summation (Δϕ23 = π) occurs. The two surviving lateral radiation lobes originate from the interferences between the two quadrupolar and the four octupolar side contributions: the lateral lobes in the lower polar plot hemisphere are suppressed by destructive interference, while the upper ones are preserved by the opposite mechanism. The quadrature sum behaves exactly as in the dipole−quadrupole combination, and also in this configuration, equal phase shifts in the summation originate a symmetric radiation pattern. In this respect it is interesting to remark that the dipole−octupole combination has not been included in the analysis, the reason being the peculiar phase relationship between the upper and lower main radiation lobes. As it is 21543

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underlined in Figure 7, upward and downward radiation fields are in phase opposition for both the dipole and the octupole: the natural consequence is that constructive and destructive interference are obtained simultaneously in the upper and lower direction, hence eliminating the possibility to obtain unidirectional radiation patterns. Figure 8c reports the radiation patterns for the coherent sum of the three lower order multipoles. This arrangement naturally produces a much richer variety of radiation patterns, and therefore, while all patterns can be understood in the adopted interference framework, we shall limit ourselves to the discussion of a few selected cases and, namely, the ones that produce the largest directional gain. It is clear that to maximize the directional gain, either in the θ = 0 or θ = π direction, the phase matching of the principal radiation lobes is needed. This consideration points at two principal multipole combinations and precisely the Δϕ12 = Δϕ23 = 0 sum, where all the upper lobes are in phase, or the Δϕ12 = π,Δϕ23 = 0 one, where the same happens for the lower lobes. The results are shown in the upper and lower left panel of Figure 8c, where upward and downward directional gain is achieved with directivity D = 7. Similar considerations may be expressed when dealing with coherent multipole sums in the m = 0 case, remembering the constraints imposed by the system polar symmetry. For our purpose it will be enough to remind the following rule of thumb: as long as electric amplitudes dominate the scattering properties, interference phenomena require m = 0 and m = ±1 unidirectional radiation to be obtained concomitantly; that is, if unidirectional emission is obtained for m = ±1 multipole combinations, the same will happen for the m = 0 case, as it is clearly illustrated in all the radiation patterns of Figure 8. Figure 9shows forward D+ = D(0,0) and backward D− = D(π,0) radiation directional gains for dipole−quadrupole and

D+, −(x) = Dn

⎛ ⎜1 ± ⎝

Dn + 1 Dn

1 + x2

⎞2 x⎟ ⎠ (5)

where x = an+1,m=± 1/an,m=± 1 corresponds to the multipole ratios; Dn = (2n+1)/2 is the pure multipole directivity; and the plus and minus signs correspond to forward and backward radiation, respectively. Complete suppression of the backward radiation occurs when the numerator of eq 5 goes to zero and is therefore obtained for x = ((Dn)/(Dn+1))1/2, while maximum forward radiation is reached at x = ((Dn+1)/(Dn))1/2 with D+(((Dn+1)/(Dn))1/2) = Dn + D n+1. In light of these considerations, it is reasonable to choose equal amplitudes for optimal multipole coherent sums since this approach corresponds to a compromise between complete suppression of backward radiation and maximum enhancement of the forward one. At last it is worth noting that the slow variation of the directional gain curves around the maxima and minima allows us to depart from optimal conditions without compromising the system unidirectional performances. It is now clear that the radiation patterns (Figures 3c and 5c) observed for our AN structures are easily interpreted in the present theoretical framework. If we first focus our attention on the smaller Rs = 75 nm nanoantenna, the interpretation of the results of Figure 3c is straightforward on account of the multipole interference model and of the relative amplitudes and phase relationships shown in Figure 3a,b. In fact the radiation patterns for top and of f dipoles (upper panels of Figure 3c) are virtually identical to those presented in Figure 8a for the Δϕ12 = π phase opposition configuration, the only difference being a slightly lower directivity attributed to minor deviations from ideal amplitude and phase values. The radiation patterns for the center configuration are also easily explained in the context of our theoretical framework. Directional radiation is simply explained by in phase and phase opposition dipole−quadrupole combinations, where the lower directivity and weaker unidirectional nature are a consequence of the worse phase matching and a greater unbalance in the multipole amplitudes. It is nevertheless remarkable that, as suggested by the results of Figure 9, conditions so far away from the optimum are able to produce directional emission and color redirection at the subwavelength scale. The radiative pattern results for the larger Rs = 200 nm AN are also easily fitted in the multipole interference theoretical framework. The patterns shown in the upper panels of Figure 5c for top and of f configurations evidently resemble the one reported in the third panel of Figure 8b, corresponding to the phase opposition quadrupole−octupole combination. The small discrepancies in terms of directivity and shape are again ascribed to slightly nonideal amplitude and phase matching, as for the Rs = 75 nm nanoshell. The results reported in the lower panels of Figure 5c (top, λ = 400 nm; center, λ = 680 nm) may also be understood it terms of coherent in phase and phase opposition multipole sums; nevertheless, the discrepancies against the ideal values of Figure 8 are only partially explained by nonoptimal amplitude and phase relationships, as can be readily understood by observing the m = 0 radiation patterns. It is evident that the rule of thumb that ties the m = 0 and m= ± 1 unidirectional properties does not hold true in the present situation, as a result of non negligible bnm magnetic coefficients originated by the AN larger sizes. Still, this extremely simple model is able to capture and predict all the essential features of

Figure 9. Forward (D+) and backward (D−) directional gains (eq 4) as a function of the multipole ratio x = an+1,m=±1/an,m=±1 for n = 1,2. Black lines indicate the dipole−quadrupole combination, and blue lines indicate the quadrupole−octupole combination.

quadrupole−octupole combinations, as a function of the multipole amplitude ratios. As usual, we shall limit ourselves to the discussion of the m = ±1 case, fixing the phase relationship between the multipoles to Δϕ12 = 0 and Δϕ23 = 0; that is, the multipoles are combined in phase. It is straightforward to show that the all the resulting directional gain curves are described by the following simple formula 21544

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ACKNOWLEDGMENTS This work has been partially supported by the Progetto di Ateneo CPDA101587 and the Strategic Project PLATFORMS STPD089KSC of the University of Padova, Italy

the complex radiation patterns emerging from the studied AN structures. Finally, we wish to underline an extremely intriguing feature of the asymmetric plasmonic nanoshells and namely the fact that larger FB directional gains and emission enhancements are obtained for downward radiation and for top and of f dipole configurations. Of course this implies that the energy is leaving the shell from its thicker section and from the side opposite to the source position, which is not at first an intuitive result. One therefore might wonder which kind of mechanism is able to transfer the energy from one side of the shell to the other. With no intention to fully explain the nature of this energy transport mechanism, we investigated the time evolution of the Ex component of the local field, in the case of the smaller AN 520 nm resonance, to shed some light on the present issue. Local-field intensity animations (see Supporting Information) indeed clarify the nature of the phenomenon, as they show surface waves emerging from the dipole feed and moving across the external shell surface, to finally couple to free propagating radiation at the θ = π point. To explain in detail the intimate nature of these surface waves is of course outside the scope of the present article, and is therefore left as future work, but we nevertheless speculate that the observed behavior might be included in the framework of our multipole interference approach. In summary, we proposed AN as directional subwavelength nanoantennas and investigated their unidirectional emission properties. It has been demonstrated that ANs are capable of 2 orders of magnitude directional emission enhancements and of forward−backward directional gains as large as 30 dB for Rs = 75 nm asymmetric plasmonic nanoshells. It has also been established that ANs are able to support unidirectional color routing for different sizes and geometries and for different emitter positions. It was then shown that all the results may be explained in an extremely simple theoretical framework which lays its foundation on the interference phenomena occurring in coherent multipole sums. The proposed model may of course be extended to arbitrary geometries, given that a method to calculate the mode amplitudes of the arising charge distributions is provided. It is finally worth noting that ANs are easily synthesized on large scales with standard colloidal chemistry techniques and may be employed as independent, intrinsically unidirectional nanoemitters if the local light sources are embedded in the dielectric core. The illustrated results, along with the simple proposed theoretical framework, may find important applications for light manipulation at the nanoscale, and in particular for sensing, high efficiency light emission, and light harvesting applications.

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ASSOCIATED CONTENT

* Supporting Information S

Directional emission animations for Ex intensity in the case of the Rs = 75 nm AN and top position dipole feed. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 21545

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